Abstract
A set S ⊆ V (G) is called a geodetic set if every vertex of G lies on a shortest u-v path for some u, v ∈ S, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by . A set C ⊆ V (G) is called a chromatic set if C contains all vertices of different colors in G, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by . A geo-chromatic set Sc ⊆ V (G) is both a geodetic set and a chromatic set. The geo-chromatic number of G is the minimum cardinality among all geo-chromatic sets of G. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.
Highlights
Products of structures are a fundamental construction in mathematics, for which theorems abound in set theory, category theory, universal algebra etc
A set S ⊆ V (G) is called a geodetic set if every vertex of G lies on a shortest u-v path for some u, v ∈ S, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by gn (G)
A set C ⊆ V (G) is called a chromatic set if C contains all vertices of different colors in G, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by χ (G)
Summary
Products of structures are a fundamental construction in mathematics, for which theorems abound in set theory, category theory, universal algebra etc. The most famous, well studied graph product is the cartesian product It extends many properties, and carries metric space structure with it. The depth of convexity theory enables study of geodeticity in graphs to further heights Another interesting concept in graphs that finds numerous applications is that of coloring. These two concepts are combined to give geochromatic number, which acts as a double layered measure that covers all vertices in a graph containing all color class representations. In this paper we determine the geodetic number of 2-cartesian product of some graphs and extend them to find geochromatic number.
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